Why does this radar speed sign show a decreasing speed, despite the fact that I'm accelerating? Everyday when I'm going back home from college, I pass by this big radar speed sign and I've noticed that it does a peculiar thing. The sign is placed on a downward slope of the street. So say I'm driving on 35 MPH, and I let my foot off the gas pedal completely, the street makes my car go a little faster since it's steep, and my speedometer reads 40 MPH, but the speed on the radar goes down instead of up, it gradually goes down and shows 31 or 32 MPH. 
Can anyone explain this to me? Or is the radar just faulty?
And if the radar is not faulty then which speed am I actually going?
 A: It's most likely that the radar gun is just locking on to a different vehicle.  Depending on how it's filtering and processing the received signal, it could even be measuring the speed of a car going in the opposite direction.
A: I notice the same thing on my bicycle.  As I approach a certain radar sign it says I'm going 10 mph, so I start peddling faster to see how fast I can go, but by then the sign is more off to the side instead of in front of me and the readout on the sign is decreasing!
The radar unit is probably measuring how fast I am approaching it, not my speed over the ground.  In other words, it is measuring the component of my velocity in the direction of the sign relative to my position.  If I was doing 40 mph and the sign was reading 32 mph, the angle between my velocity vector and the position vector from me to the sign would be $\arccos{(32/40)} \approx 37^{\circ}$.  If the sign was about 10 feet from my straight-line course, i.e. from the middle of the lane, I would be about $10/\tan{37^{\circ}} \approx 13.3 ft$ from the point in my path that is closest to the sign.  That is pretty close, practically passing the sign.
So to me it would be reasonable for you to see the sign reading 32 mph when you are about one car length from passing it at 40 mph.  I would say the radar sign is functioning as it was intended.  On the other hand, if you are still a block away from the sign, and see the sign reading 20% less than your speedometer, well, I could not explain that.
The following is a more technical explanation of how the radar sign determines speed.  The sign sends out a radio pulse and listens for the pulse to echo off a vehicle.  The sign measures the time between sending the pulse and receiving the echo.  That time is converted to a distance.  Then the process is repeated to get a new distance.  The new distance is different because the vehicle has moved (closer).  The calculated vehicle speed is determined by the two distances and the time between the measurements.  The send-listen-calculate process can be repeated many times per second.
The speed calculated by that procedure could be called the velocity of approach, which is not the true velocity of the vehicle.   The true velocity consists of how fast the vehicle is approaching and how fast it is veering, or going around the sign.  Even though the vehicle is moving in a straight line, the direction of the vehicle from the sign is changing.  This change in apparent direction is what I am calling veering, meaning to go clockwise around the compass.  As the vehicle gets closer to the sign, the veering effect becomes more significant, which means the calculated speed becomes less accurate.  
For example, on a north-south street the vehicle may approach the radar sign from the north.  As it nears the sign, the vehicle may be off to the northeast and then due east as it passes the sign.  However, the sign does not detect the change in direction, only the change in distance.  The sign is only sending out a pulse and waiting for an echo without regard to the direction of the echo.  A navigation radar would be able to measure the true velocity, but the radar in the sign is not that sophisticated.  
A: 
How do radar speed signs work?

To answer the question raised in the title, radar signs (and radar guns in general) measure the Doppler shift between the outgoing signal and reflected signal. This frequency shift translates directly to $\frac{dr}{dt}$, the rate of change of the distance between the acquired object and the radar. This range rate measurement is a scalar; velocity is a vector. (Another name is radial component of velocity.)
Ignoring measurement errors, the speed as measured by a radar gun is less than or equal to the vehicle's actual speed. Denoting the angle between the velocity vector of a vehicle as acquired by a radar gun and the vector from the vehicle toward the radar as $\theta$, the vehicle's speed as measured by the radar gun is $\cos\theta\;v_\text{vehicle}$.
The cosine term introduces a cosine error between the measured and actual speed. This error is small when $\theta$ is small, but it grows as the angle increases. The range rate speed becomes zero when the vehicle's velocity vector becomes orthogonal to the displacement vector between the vehicle and the radar gun.


The sign is placed on a downward slope of the street. So say I'm driving on 35 MPH, and I let my foot off the gas pedal completely, the street makes my car go a little faster since it's steep, and my speedometer reads 40 MPH, but the speed on the radar goes down instead of up, it gradually goes down and shows 31 or 32 MPH.

Radar signs aren't placed on the street. They instead are placed next to the street. This means the cosine errors become significant as you approach the sign. The downward slope of the street may also be a factor in these cosine errors.
